Grassmannian.info

A periodic table of (generalised) Grassmannians.

Orthogonal Grassmannian $\OGr(2,15)$

Basic information
dimension
23
index
12
Euler characteristic
84
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 16 } = 5$, $\mathrm{b}_{ 18 } = 5$, $\mathrm{b}_{ 20 } = 6$, $\mathrm{b}_{ 22 } = 6$, $\mathrm{b}_{ 24 } = 6$, $\mathrm{b}_{ 26 } = 6$, $\mathrm{b}_{ 28 } = 5$, $\mathrm{b}_{ 30 } = 5$, $\mathrm{b}_{ 32 } = 4$, $\mathrm{b}_{ 34 } = 4$, $\mathrm{b}_{ 36 } = 3$, $\mathrm{b}_{ 38 } = 3$, $\mathrm{b}_{ 40 } = 2$, $\mathrm{b}_{ 42 } = 2$, $\mathrm{b}_{ 44 } = 1$, $\mathrm{b}_{ 46 } = 1$
$\mathrm{Aut}^0(\OGr(2,15))$
$\mathrm{SO}_{ 15 }$
$\pi_0\mathrm{Aut}(\OGr(2,15))$
$1$
$\dim\mathrm{Aut}^0(\OGr(2,15))$
105
Projective geometry
minimal embedding

$\OGr(2,15)\hookrightarrow\mathbb{P}^{ 104 }$

degree
832048
Hilbert series
1, 105, 4080, 88179, 1270815, 13537524, 113859200, 791224200, 4695002676, 24385860460, 113015849856, 474507071220, 1827174287820, 6518164602264, 21722800137600, 68110029598100, 202120945079625, 570619024282125, 1539370437368400, 3983638152414375, ...
Exceptional collections
• Kuznetsov constructed a full exceptional sequence in 2008, see MR2434094.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\OGr(2,15))$ are given by:

Homological projective duality